Conditional κ = 7.10465204542247e-4: fixed prime, seven finite levels and η = 10⁻²⁴ on #254

[hvaliconAI]

Conditional κ = 710465204542247142217/10²⁴ = 7.10465204542247142217 × 10⁻⁴, an exact gain of 3909417142217/10^24 (≈ 3.909 × 10⁻¹²) over #254 (71046520063283/10^17). The bit supplier binds.

This is an arithmetic refinement of Dugongue's #254. The physical construction is unchanged: no word, frame, bank, chart or finite cost differs. The method is gabriele-nespoli's #235, the same one #243, #247, #248, #252 and #253 applied to other parents. No record beyond the parent's physics is claimed.

What changes

• Fixed prime. q = 2¹²⁷ − 1 is fixed before every width. The verifier recomputes all 125 Lucas–Lehmer updates; Conditional κ = 7.08582414684555347682e-4: fixed prime and finite leaves on PR237 banks #235 also kernel-checks the prime in Lean.
• Rare density. The retained bit theorem gives the bound 2m³/q, so ρ = 3456000/q < 10⁻¹⁶ replaces the envelope 10⁻¹⁶. Every rare edge still pays the full 32m² fallback.
• Ordinary levels. Seven acyclic completed levels replace three. All four admission gaps are positive at every level.
• Assembly. η = 10⁻²⁴, as in Conditional κ = 7.09424410568126e-4: fixed prime, seven finite levels and η = 10⁻²⁴ on #240's five-stage banks #243 and the reviewed main result, with β = 10⁻⁹. The outer-47 assembly is unchanged, on a 10⁻²⁴ grid.

Checks (python3 -B research/fixed-prime-pr254/verify.py, standard library only)

• Parent reproduced. The verifier recovers Response-kernel entrance gauges: 132 helper-pair rewrites, κ = 7.1046520063283e-4 #254's coarse saving 71097032054791/10^17 and its κ 71046520063283/10^17 exactly. It uses Response-kernel entrance gauges: 132 helper-pair rewrites, κ = 7.1046520063283e-4 #254's own receipts, pinned by SHA-256, and Response-kernel entrance gauges: 132 helper-pair rewrites, κ = 7.1046520063283e-4 #254's parameters.
• Two independent brackets. The public interval engine and an independent atanh engine bracket the new coarse saving on a 10⁻²⁷ grid. The old 10⁻¹⁶ density fails at the new point, so the gain depends on the proved density.
• Outer assembly. All 47 strict constraints and seven margins pass, and the adjacent κ grid point is rejected.
• Not a rounding change. The new κ exceeds Response-kernel entrance gauges: 132 helper-pair rewrites, κ = 7.1046520063283e-4 #254's unrounded outer bound.
• Finite bill below q. Response-kernel entrance gauges: 132 helper-pair rewrites, κ = 7.1046520063283e-4 #254's invoice charges its 40 literal replicas and its full bank stock. Its coefficient 62447396383377201 (< 2⁵⁶) and its 95-bit payload prefix both stay below 2⁸⁰ < q.

Local replay

I ran #254's own verifier on Linux (g++ with Boost 1.86 headers) and all seven native checkers passed. Its fresh COHORT-EXACT-PRICE.json and COHORT-FINITE-INVOICE.json are byte-identical to #254's packaged expected/ receipts. --parent-output DIR accepts such a fresh run directly. The certificate is byte-identical whether generated on Linux or on Windows.

Scope

The result is conditional on every interface #254 retains: the all-size compiler, common weighted chart, restored rows, selectors, routing, prime supply, precision/recovery, complex symbolic correctness and the analytic reduction. The complex profile and finite bridge are vendored from #251's fresh certificate (#234's five-stage complex supplier) and re-checked here. No unconditional theorem, Lean certificate of the bound, or runtime claim is made.

The branch is stacked on #254 (9ce32ef), so the diff against main also contains #254's package. The new files are only those under research/fixed-prime-pr254/.

Credits: Dugongue (#254), eumemic (#249 and the source527 lineage), gabriele-nespoli (#235), Rohan Arun (#237, #243), hcg890 and Jacob Sussman (#234), Evan McKinney (#197). Prepared with Claude (Anthropic) assistance.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Thank you @hvaliconAI. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Fixed-prime seven-level arithmetic on PR254.

Review and limits: Fresh fixed-prime seven-level verifier passed, with the actual parent finite invoice and fresh PR254 native replay. Prime constraints, arithmetic and source bindings are distinct checks.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 3ba1c146c523c0305fe5f1a7bc4d3c5aebb9192c. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.